True. Fans are the worst judges there are. That being said, judging a national team coach is quite difficult because the number of games is limited, unlike a club where they can see the progress every week or every transfer window. A national team needs to produce results befitting their image in most competitions. Then again, that's the job of a functioning sports association. The bad news is we don't even know if the guys running the association are competent or not, most of the time (at least in my country).
I think that a national team should be managed by a more experienced coach, especially one who has already proved the ability to win titles at club level. But if you look at Gattuso's rather modest coaching career, there is nothing remarkable about it. So I can easily assume that the national team under Gattuso could be even worse than the one under Spalletti. Of course, these are just my assumptions. I have always supported the Italy national team. I have always liked them as a team, but lately they have been going through tough times, and I do not think it will end with Gattuso.